Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

March 28, 2024

Why Mathematics is Boring

Posted by John Baez

I’m writing a short article with some thoughts on how to write math papers, with a provocative title. It’s due very soon, so if you have any thoughts about this draft I’d like to hear them soon!

Posted at 10:21 PM UTC | Permalink | Followups (36)

March 23, 2024

Counting Points on Elliptic Curves (Part 3)

Posted by John Baez

In Part 1 of this little series I showed you Wikipedia’s current definition of the LL-function of an elliptic curve, and you were supposed to shudder in horror. In this definition the LL-function is a product over all primes pp. But what do we multiply in this product? There are 4 different cases, each with its own weird and unmotivated formula!

In Part 2 we studied the 4 cases. They correspond to 4 things that can happen when we look at our elliptic curve over the finite field 𝔽 p\mathbb{F}_{p}: it can stay smooth, or it can become singular in 3 different ways. In each case we got a formula for number of points the resulting curve over the fields 𝔽 p k\mathbb{F}_{p^k}.

Now I’ll give a much better definition of the LL-function of an elliptic curve. Using our work from last time, I’ll show that it’s equivalent to the horrible definition on Wikipedia. And eventually I may get up the nerve to improve the Wikipedia definition. Then future generations will wonder what I was complaining about.

Posted at 1:00 AM UTC | Permalink | Followups (7)

March 13, 2024

Counting Points on Elliptic Curves (Part 2)

Posted by John Baez

Last time I explained three ways that good curves can go bad. We start with an equation like

y 2=P(x) y^2 = P(x)

where PP is a cubic with integer coefficients. This may define a perfectly nice smooth curve over the complex numbers — called an ‘elliptic curve’ — and yet when we look at its solutions in finite fields, the resulting curves over those finite fields may fail to be smooth. And they can do it in three ways.

Let’s look at examples.

Posted at 8:00 PM UTC | Permalink | Followups (10)

March 10, 2024

Counting Points on Elliptic Curves (Part 1)

Posted by John Baez

You’ve probably heard that there are a lot of deep conjectures about LL-functions. For example, there’s the Langlands program. And I guess the Riemann Hypothesis counts too, because the Riemann zeta function is the grand-daddy of all LL-functions. But there’s also a million-dollar prize for proving the Birch-Swinnerton–Dyer conjecture about LL-functions of elliptic curves. So if you want to learn about this stuff, you may try to learn the definition of an LL-function of an elliptic curve.

But in many expository accounts you’ll meet a big roadblock to understanding.

The LL-function of elliptic curve is often written as a product over primes. For most primes the factor in this product looks pretty unpleasant… but worse, for a certain finite set of ‘bad’ primes the factor looks completely different, in one of 3 different ways. Many authors don’t explain why the LL-function has this complicated appearance. Others say that tweaks must be made for bad primes to make sure the LL-function is a modular form, and leave it at that.

I don’t think it needs to be this way.

Posted at 9:08 PM UTC | Permalink | Followups (4)

March 9, 2024

Semi-Simplicial Types, Part I: Motivation and History

Posted by Mike Shulman

(Jointly written by Astra Kolomatskaia and Mike Shulman)

This is part one of a three-part series of expository posts on our paper Displayed Type Theory and Semi-Simplicial Types. In this part, we motivate the problem of constructing SSTs and recap its history.

Posted at 5:33 PM UTC | Permalink | Followups (2)

March 3, 2024

Modular Curves and Monstrous Moonshine

Posted by John Baez

Recently James Dolan and I have been playing around with modular curves — more specifically the curves X 0(n)X_0(n) and X 0 +(n)X^+_0(n), which I’ll explain below. Monstrous Moonshine says that when pp is prime, the curve X 0 +(p)X^+_0(p) has genus zero iff pp divides the order of the Monster group, namely

p=2,3,5,7,11,13,17,19,23,29,31,41,47,59,71 p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71

Just for fun we’ve been looking at n=11n = 11, among other cases. We used dessins d’enfant to draw a picture of X 0(11)X_0(11), which seems to have genus 11, so for X 0 +(11)X^+_0(11) to have genus zero it seems we want the picture for X 0(11)X_0(11) to have a visible two-fold symmetry. After all, the torus is a two-fold branched cover of the sphere, as shown by Greg Egan here:

But we’re not seeing that two-fold symmetry. So maybe we’re making some mistake!

Maybe you can help us, or maybe you’d just like a quick explanation of what we’re messing around with.

Posted at 9:43 PM UTC | Permalink | Followups (4)